Question

``In the coordinate plane the point X (0,-3) is translated to the point X'(-3,0). Under the same translation the points Y(4,-6) and the Z(-4,-5) as translated to Y' and Z'respectively what are the coordinates of Y' and Z'?''.

Answer 1

Question#37457 - Mathematics - Geometry

In the coordinate plane the point $X(\theta, -3)$ is translated to the point $X'(-3, 0)$. Under the same translation the points $Y(4, -6)$ and the $Z(-4, -5)$ as translated to $Y'$ and $Z'$ respectively what are the coordinates of $Y'$ and $Z'$?

Solution:

Point $X(x_{x}, y_{x})$ is translated using the rule $(x_{x}, y_{x}) \to (x_{x} + \Delta x, y_{x} + \Delta y)$

Given:

$$x _ {x} = 0, \quad y _ {x} = - 3$$ $$x _ {x} + \Delta x = x _ {x} ^ {\prime} = - 3, \quad y _ {x} + \Delta y = y _ {x} ^ {\prime} = 0$$

So

$$0 + \Delta x = - 3, \quad - 3 + \Delta y = 0$$ $$\Delta x = - 3, \quad \Delta y = 3$$

If the points $Y(x_{y}, y_{y})$ and the $Z(x_{x}, y_{x})$ translated under the same translation, then

$$x _ {y} ^ {\prime} = x _ {y} + \Delta x = 4 - 3 = 1$$ $$y _ {y} ^ {\prime} = y _ {y} + \Delta y = - 6 + 3 = 3$$ $$x _ {z} ^ {\prime} = x _ {z} + \Delta x = - 4 - 3 = - 7$$ $$y _ {z} ^ {\prime} = y _ {z} + \Delta y = - 5 + 3 = - 2$$

Given:

$$x _ {y} = 4, y _ {y} = - 6 \text{ and } x _ {z} = - 4, y _ {z} = - 5, \text{ so}$$ $$x _ {y} ^ {\prime} = 4 - 3 = 1$$ $$y _ {y} ^ {\prime} = - 6 + 3 = 3$$

and

$$x _ {z} ^ {\prime} = - 4 - 3 = - 7$$ $$y _ {z} ^ {\prime} = - 5 + 3 = - 2$$

Answer: $Y' = (1, 3)$ and $Z'(-7, -2)$